Mumford-Tate groups of 1-motives and Weil pairing

TL;DR

This paper explores how the geometry of 1-motives influences their motivic Galois groups and Mumford-Tate groups, providing classifications and computations that shed light on period relations and Grothendieck's conjecture.

Contribution

It establishes a geometric framework linking 1-motives to their Mumford-Tate groups and classifies period relations, advancing understanding of motivic Galois groups.

Findings

Dimension of motivic Galois group determined by 1-motive geometry

Explicit description of Mumford-Tate group action on period matrices

Classification of polynomial relations between periods in semi-elliptic case

Abstract

We show how the geometry of a 1-motive $M$ (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group ${\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}(M)$. Fixing periods matrices $\Pi_M$ and $\Pi_{M^*}$ associated respectively to a 1-motive $M$ and to its Cartier dual $M^*,$ we describe the action of the Mumford-Tate group of $M$ on these matrices. In the semi-elliptic case, according to the geometry of $M$ we classify polynomial relations between the periods of $M$ and we compute exhaustively the matrices representing the Mumford-Tate group of $M$. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives.